Finite Group Databasebeta

← #44560414

Subgroups — #44560414

Order 32128.

454 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632221C211yesmaximal, characteristic, complemented
2C2×C4×D1004C211yesmaximal, characteristic, complemented
3#30632023C211yesmaximal, characteristic, complemented
4#30632006C211yesmaximal, characteristic, complemented
5#30631988C212yesmaximal, complemented
6#30631903C212yesmaximal, complemented
7#30631840C211yesmaximal, characteristic, complemented
8#30631823C211yesmaximal, characteristic
9C4×#16860799C212yesmaximal, complemented
10C4×D2008C212yesmaximal, complemented
11#431×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C4×D251—21no—
14C2×#16860799C2212yes—
15C2×D2008C2212yes—
16#22548709—21no—
17#22548707C2211yescharacteristic, complemented
18#22548703C2211yescharacteristic, complemented
19#22548698—21no—
20#22548694C2211yescharacteristic
21#22548689C2211yescharacteristic
22C4×Dic502C2211yescharacteristic, complemented
23C4×D1004—21no—
24C4×D1004—21no—
25C4×D1004C2212yes—
26C4×D1004C2212yes—
27C4×D1004C2211yescharacteristic, complemented
28C4×C22⋊D251—21no—
29#22548672—21no—
30C2×C4⋊Dic251C2211yescharacteristic
31C2×(C2×D251)⋊C4—21no—
32#22548649C2212yescomplemented
33#22548647C2212yescomplemented
34#22548640C2212yescomplemented
35#22548637C2212yes—
36#22548637C2212yes—
37#22548637C2212yescomplemented
38C2×C42×C251C2211yescharacteristic
39C4⋊C8×C251C2212yescomplemented
40C22⋊C8×C251C2211yescharacteristic
41C22⋊C8×C251C2211yescharacteristic
42C4×C8×C251C2212yescomplemented
43C23×D251—21no—
44C2×Dic502C2311yescharacteristic
45C2×D1004—24no—
46C2×D1004—21no—
47C2×D1004C2311yescharacteristic
48C2×D1004C2312yes—
49C2×C22⋊D251—21no—
50C22×Dic251—21no—
51C2×C4×D251—22no—
52C2×C4×D251—22no—
53C2×C4×D251—21no—
54C2×C4×D251—44no—
55#16860799—24no—
56D2008—24no—
57C4⋊Dic251—21no—
58C4⋊Dic251C2311yescharacteristic
59C4⋊Dic251C2311yescharacteristic
60C4⋊Dic251C2312yes—
61Dic251⋊C4—21no—
62Dic251⋊C4—21no—
63Dic251⋊C4—21no—
64Dic251⋊C4—21no—
65C22⋊Dic251—21no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—22no—
68(C2×D251)⋊C4—21no—
69(C2×D251)⋊C4—21no—
70(C2×D251)⋊C4—21no—
71(C2×D251)⋊C4—21no—
72C4×Dic251—21no—
73C22×C4×C251D411yescharacteristic
74C22×C4×C251D411yescharacteristic
75C22×C4×C251C2311yescharacteristic
76C2×C8×C251C2312yes—
77C2×C8×C251C2312yes—
78C42×C251D411yescharacteristic
79C42×C251D411yescharacteristic
80C42×C251C2311yescharacteristic
81C42×C251C2311yescharacteristic
82C22×D251—24no—
83C22×D251—22no—
84C22×D251—21no—
85C22×D251—42no—
86C22×D251—42no—
87Dic502—22no—
88D1004—24no—
89D1004—22no—
90D1004—22no—
91D1004#4114yes—
92C22⋊D251—42no—
93C2×Dic251—21no—
94C2×Dic251—21no—
95C2×Dic251—22no—
96C2×Dic251—21no—
97C2×Dic251—42no—
98C4×D251—44no—
99C4×D251—42no—
100C4×D251—44no—
101C23×C251C2×D411yescharacteristic
102C2×C4×C251—22no—
103C2×C4×C251—22no—
104C2×C4×C251—22no—
105C2×C4×C251C2×D411yescharacteristic
106C2×C4×C251C2×D411yescharacteristic
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2×D411yescharacteristic
110C2×C4×C251C2411yescharacteristic
111C8×C251—22no—
112C8×C251—22no—
113C22×C251—21no—
114C22×C251—21no—
115C22×C251#13512yes—
116C22×C251C22×D411yescharacteristic
117C2×D251—24no—
118C2×D251—48no—
119C2×D251—42no—
120C2×D251—42no—
121C2×D251—41no—
122C2×D251—41no—
123C2×D251—41no—
124C4×C251—21no—
125C4×C251—21no—
126C4×C251—21no—
127C4×C251—21no—
128C4×C251AGL(1,ℤ/8)12yes—
129C4×C251C2×#4112yes—
130Dic251—41no—
131Dic251—41no—
132Dic251—42no—
133Dic251—41no—
134C2×C251—22no—
135C2×C251#52511yescharacteristic
136C2×C251C2×#13511yescharacteristic
137C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
138D251—44no—
139D251—81no—
140C251#310511yescharacteristic, complemented, Sylow
141#3105—2511nomaximal, Sylow
142#529—2511no—
143C2×C4×D4—2511no—
144#484—2511no—
145#481—2511no—
146#478—2512no—
147#463—2512no—
148#450—2511no—
149#447—2511no—
150C4×SD16—2512no—
151C4×D8—2512no—
152#431D25111yescharacteristic, complemented
153C22×D4—2511no—
154C23×C4—5021no—
155C2×SD16—2512no—
156C2×D8—2512no—
157#126—5021no—
158#125—2511no—
159#124—2511no—
160#123—5021no—
161C22⋊Q8—2511no—
162#121—2511no—
163C4×Q8—2511no—
164C4×D4—5021no—
165C4×D4—5021no—
166C4×D4—5021no—
167C4×D4—2512no—
168C4×D4—2512no—
169C4×D4—2511no—
170#117—5021no—
171C2×C4⋊C4—2511no—
172C2×C22⋊C4—5021no—
173C2×C42C2×D25111yescharacteristic
174#107—2512no—
175#106—2512no—
176C4⋊C8C2×D25112yescomplemented
177#103—2512no—
178#102—2512no—
179#102—2512no—
180#102—2512no—
181C22⋊C8C2×D25111yescharacteristic
182C22⋊C8C2×D25111yescharacteristic
183C4×C8C2×D25112yescomplemented
184C24—5021no—
185C2×Q8—2511no—
186C2×D4—5024no—
187C2×D4—5021no—
188C2×D4—5021no—
189C2×D4—2511no—
190C2×D4—2512no—
191C22×C4—10044no—
192C22×C4—5021no—
193C22×C4—5022no—
194C22×C4—5021no—
195C22×C4—5022no—
196C22×C4D100411yescharacteristic
197C22×C4D100411yescharacteristic
198C22×C4C22×D25111yescharacteristic
199SD16—5024no—
200D8—5024no—
201C2×C8C22×D25112yes—
202C2×C8C22×D25112yes—
203C4⋊C4—5021no—
204C4⋊C4—5021no—
205C4⋊C4—5021no—
206C4⋊C4—5021no—
207C4⋊C4—5021no—
208C4⋊C4—2511no—
209C4⋊C4—2511no—
210C4⋊C4—2512no—
211C22⋊C4—5021no—
212C22⋊C4—5022no—
213C22⋊C4—5022no—
214C22⋊C4—5021no—
215C22⋊C4—5021no—
216C22⋊C4—5021no—
217C22⋊C4—5021no—
218C42—5021no—
219C42D100411yescharacteristic
220C42D100411yescharacteristic
221C42C22×D25111yescharacteristic
222C42C22×D25111yescharacteristic
223C23—10042no—
224C23—10042no—
225C23—5024no—
226C23—5021no—
227C23—5022no—
228C23C2×D100411yescharacteristic
229Q8—5022no—
230D4—10042no—
231D4—5024no—
232D4—5022no—
233D4—5022no—
234D4—2514no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—10044no—
239C2×C4—10042no—
240C2×C4—10044no—
241C2×C4—10042no—
242C2×C4—5021no—
243C2×C4—5021no—
244C2×C4—5021no—
245C2×C4—5022no—
246C2×C4C2×D100411yescharacteristic
247C2×C4C2×D100411yescharacteristic
248C2×C4C2×D100411yescharacteristic
249C2×C4C2×D100411yescharacteristic
250C2×C4C2×D100411yescharacteristic
251C2×C4C23×D25111yescharacteristic
252C8—22no—
253C8—22no—
254C22—21no—
255C22—21no—
256C22—10048no—
257C22—10041no—
258C22—10041no—
259C22—10041no—
260C22—10042no—
261C22—10042no—
262C22—5024no—
263C22#2254873712yes—
264C22C22×D100411yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—10041no—
270C4—10041no—
271C4—10042no—
272C4—10041no—
273C4#2254874312yes—
274C4#41×D25112yes—
275C2—22no—
276C2—20081no—
277C2—10044no—
278C2#3063219911yescharacteristic
279C2C2×#2254873711yescharacteristic
280C2C2×#2254874311yescharacteristic
subgroupG/H
centerC2×C4C2×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#431×C251C2
socleC22×C251C22×D4
prime radical#30631823C2
hypercenter#431D251
hypocenterC251#3105
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #431

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #30632221C2semidirect
2. C2×C4×D1004C2semidirect
3. #30632023C2semidirect
4. #30632006C2semidirect
5. #30631988C2semidirect
6. #30631903C2semidirect
7. #30631840C2semidirect
9. C4×#16860799C2semidirect
10. C4×D2008C2semidirect
11. #431×C251C2semidirect
17. #22548707C22semidirect
18. #22548703C22semidirect
22. C4×Dic502C22semidirect
27. C4×D1004C22semidirect
32. #22548649C22semidirect
33. #22548647C22semidirect
34. #22548640C22semidirect
37. #22548637C22semidirect
39. C4⋊C8×C251C22semidirect
42. C4×C8×C251C22semidirect
140. C251#3105semidirect
152. #431D251semidirect
176. C4⋊C8C2×D251semidirect
183. C4×C8C2×D251semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#306322212
C2×C4×D10042
#306320232
#306320062
#306319882
#306319032
#306318402
#306318232
C4×#168607992
C4×D20082
#431×C2512
#3105251